Stella, a computer program available in three versions (Great Stella, Small Stella and Stella4D), was created by Robert Webb of Australia. The programs contain a large library of polyhedra which can be manipulated and altered in various ways.

In 2007, a Stella4D version was added, allowing the generation and display of four-dimensional polytopes (polychora), including a library of all convex uniform polychora, and all currently known nonconvex star polychora, as well as the uniform duals. They can be selected from a library or generated from user created polyhedral vertex figure files.

Stella provides a configurable workspace comprising several panels. Once a model has been selected from the range available, different views of it may be displayed in each panel. These views can also include measurements, symmetries and unfolded nets.

A variety of operations may be performed on any polyhedron. In 3D these include: stellation, faceting, augmentation, excavation, drilling and dualising.

Other features include spring network relaxation, generation of the convex hull, and generation of cupolaic blends and related figures.

1.
Computer program
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A computer program is a collection of instructions that performs a specific task when executed by a computer. A computer requires programs to function, and typically executes the programs instructions in a processing unit. A computer program is written by a computer programmer in a programming language. From the program in its form of source code, a compiler can derive machine code—a form consisting of instructions that the computer can directly execute. Alternatively, a program may be executed with the aid of an interpreter. A part of a program that performs a well-defined task is known as an algorithm. A collection of programs, libraries and related data are referred to as software. Computer programs may be categorized along functional lines, such as software or system software. The earliest programmable machines preceded the invention of the digital computer, in 1801, Joseph-Marie Jacquard devised a loom that would weave a pattern by following a series of perforated cards. Patterns could be weaved and repeated by arranging the cards, in 1837, Charles Babbage was inspired by Jacquards loom to attempt to build the Analytical Engine. The names of the components of the device were borrowed from the textile industry. In the textile industry, yarn was brought from the store to be milled, the device would have had a store—memory to hold 1,000 numbers of 40 decimal digits each. Numbers from the store would then have then transferred to the mill. It was programmed using two sets of perforated cards—one to direct the operation and the other for the input variables, however, after more than 17,000 pounds of the British governments money, the thousands of cogged wheels and gears never fully worked together. During a nine-month period in 1842–43, Ada Lovelace translated the memoir of Italian mathematician Luigi Menabrea, the memoir covered the Analytical Engine. The translation contained Note G which completely detailed a method for calculating Bernoulli numbers using the Analytical Engine and this note is recognized by some historians as the worlds first written computer program. In 1936, Alan Turing introduced the Universal Turing machine—a theoretical device that can model every computation that can be performed on a Turing complete computing machine and it is a finite-state machine that has an infinitely long read/write tape. The machine can move the back and forth, changing its contents as it performs an algorithm

2.
Australia
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Australia, officially the Commonwealth of Australia, is a country comprising the mainland of the Australian continent, the island of Tasmania and numerous smaller islands. It is the worlds sixth-largest country by total area, the neighbouring countries are Papua New Guinea, Indonesia and East Timor to the north, the Solomon Islands and Vanuatu to the north-east, and New Zealand to the south-east. Australias capital is Canberra, and its largest urban area is Sydney, for about 50,000 years before the first British settlement in the late 18th century, Australia was inhabited by indigenous Australians, who spoke languages classifiable into roughly 250 groups. The population grew steadily in subsequent decades, and by the 1850s most of the continent had been explored, on 1 January 1901, the six colonies federated, forming the Commonwealth of Australia. Australia has since maintained a liberal democratic political system that functions as a federal parliamentary constitutional monarchy comprising six states. The population of 24 million is highly urbanised and heavily concentrated on the eastern seaboard, Australia has the worlds 13th-largest economy and ninth-highest per capita income. With the second-highest human development index globally, the country highly in quality of life, health, education, economic freedom. The name Australia is derived from the Latin Terra Australis a name used for putative lands in the southern hemisphere since ancient times, the Dutch adjectival form Australische was used in a Dutch book in Batavia in 1638, to refer to the newly discovered lands to the south. On 12 December 1817, Macquarie recommended to the Colonial Office that it be formally adopted, in 1824, the Admiralty agreed that the continent should be known officially as Australia. The first official published use of the term Australia came with the 1830 publication of The Australia Directory and these first inhabitants may have been ancestors of modern Indigenous Australians. The Torres Strait Islanders, ethnically Melanesian, were originally horticulturists, the northern coasts and waters of Australia were visited sporadically by fishermen from Maritime Southeast Asia. The first recorded European sighting of the Australian mainland, and the first recorded European landfall on the Australian continent, are attributed to the Dutch. The first ship and crew to chart the Australian coast and meet with Aboriginal people was the Duyfken captained by Dutch navigator, Willem Janszoon. He sighted the coast of Cape York Peninsula in early 1606, the Dutch charted the whole of the western and northern coastlines and named the island continent New Holland during the 17th century, but made no attempt at settlement. William Dampier, an English explorer and privateer, landed on the north-west coast of New Holland in 1688, in 1770, James Cook sailed along and mapped the east coast, which he named New South Wales and claimed for Great Britain. The first settlement led to the foundation of Sydney, and the exploration, a British settlement was established in Van Diemens Land, now known as Tasmania, in 1803, and it became a separate colony in 1825. The United Kingdom formally claimed the part of Western Australia in 1828. Separate colonies were carved from parts of New South Wales, South Australia in 1836, Victoria in 1851, the Northern Territory was founded in 1911 when it was excised from South Australia

3.
Platonic solid
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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the number of faces meeting at each vertex. Five solids meet those criteria, Geometers have studied the mathematical beauty and they are named for the ancient Greek philosopher Plato who theorized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, dice go back to the dawn of civilization with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively, some sources credit Pythagoras with their discovery. In any case, Theaetetus gave a description of all five. The Platonic solids are prominent in the philosophy of Plato, their namesake, Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, there was intuitive justification for these associations, the heat of fire feels sharp and stabbing. Air is made of the octahedron, its components are so smooth that one can barely feel it. Water, the icosahedron, flows out of hand when picked up. By contrast, a highly nonspherical solid, the hexahedron represents earth and these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cubes being the regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks. the god used for arranging the constellations on the whole heaven. Aristotle added an element, aithēr and postulated that the heavens were made of this element. Euclid completely mathematically described the Platonic solids in the Elements, the last book of which is devoted to their properties, propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the sphere to the edge length. In Proposition 18 he argues there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the goal of the deductive system canonized in the Elements

4.
Archimedean solid
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In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedrons composed of regular meeting in identical vertices, excluding the 5 Platonic solids. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices, identical vertices means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other. Excluding these two families, there are 13 Archimedean solids. All the Archimedan solids can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry, the Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra, kepler may have also found the elongated square gyrobicupola, at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a configuration of means that a square, hexagon. Some definitions of semiregular polyhedron include one more figure, the square gyrobicupola or pseudo-rhombicuboctahedron. The number of vertices is 720° divided by the angle defect. The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular, the duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices, the snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form and right-handed form. When something comes in forms which are each others three-dimensional mirror image. The different Archimedean and Platonic solids can be related to each other using a handful of general constructions, starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated, different Platonic and Archimedean solids can be created, expansion or cantellation involves moving each face away from the center and taking the convex hull. Expansion with twisting also involves rotating the faces, thus breaking the rectangles corresponding to edges into triangles, the last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as truncation of corners and edges, note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron

5.
Uniform polyhedron
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A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, Uniform polyhedra may be regular, quasi-regular or semi-regular. The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra, there are two infinite classes of uniform polyhedra together with 75 others. Dual polyhedra to uniform polyhedra are face-transitive and have regular vertex figures, the dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid. The concept of uniform polyhedron is a case of the concept of uniform polytope. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces, by a polygon they implicitly mean a polygon in 3-dimensional Euclidean space, these are allowed to be non-convex and to intersect each other. There are some generalizations of the concept of a uniform polyhedron, if the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate and these require a more general definition of polyhedra. Some of the ways they can be degenerate are as follows, some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra, some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron, there double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra, there are several polyhedra with doubled faces produced by Wythoffs construction. Most authors do not allow doubled faces and remove them as part of the construction, skillings figure has the property that it has double edges but its faces cannot be written as a union of two uniform polyhedra. Regular convex polyhedra, The Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Plato, Theaetetus, Timaeus of Locri, the Etruscans discovered the regular dodecahedron before 500 BC. Nonregular uniform convex polyhedra, The cuboctahedron was known by Plato, Archimedes discovered all of the 13 Archimedean solids. His original book on the subject was lost, but Pappus of Alexandria mentioned Archimedes listed 13 polyhedra, piero della Francesca rediscovered the five truncation of the Platonic solids, truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron. Luca Pacioli republished Francescas work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, which was drawn by Leonardo da Vinci. Johannes Kepler was the first to publish the complete list of Archimedean solids, in 1619, regular star polyhedra, Kepler discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot discovered the other two

6.
Compound of five tetrahedra
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The compound of five tetrahedra is one of the five regular polyhedral compounds. This compound polyhedron is also a stellation of the regular icosahedron and it was first described by Edmund Hess in 1876. It can be seen as a faceting of a regular dodecahedron and it can be constructed by arranging five tetrahedra in rotational icosahedral symmetry, as colored in the upper right model. It is one of five regular compounds which can be constructed from identical Platonic solids and it shares the same vertex arrangement as a regular dodecahedron. There are two forms of this compound polyhedron. Both forms together create the reflection symmetric compound of ten tetrahedra and it can also be obtained by stellating the icosahedron, and is given as Wenninger model index 24. It is a faceting of a dodecahedron, as shown at left, the compound of five tetrahedra is a geometric illustration of the notion of orbits and stabilizers, as follows. This compound is unusual, in that the figure is the enantiomorph of the original. If the faces are twisted to the right, then the vertices are twisted to the left, when we dualise, the faces dualise to right-twisted vertices and the vertices dualise to left-twisted faces, giving the chiral twin. Figures with this property are extremely rare, Compound of ten tetrahedra Wenninger, Magnus. Metal Sculpture of Five Tetrahedra Compound VRML model, Compounds of 5 and 10 Tetrahedra by Sándor Kabai, The Wolfram Demonstrations Project

7.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra

8.
Dual polyhedron
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron, duality preserves the symmetries of a polyhedron. Therefore, for classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. There are many kinds of duality, the kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. The duality of polyhedra is often defined in terms of polar reciprocation about a concentric sphere. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2. The vertices of the dual are the reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required plane at infinity. Some theorists prefer to stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged

9.
Vertex figure
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In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Take some vertex of a polyhedron, mark a point somewhere along each connected edge. Draw lines across the faces, joining adjacent points. When done, these form a complete circuit, i. e. a polygon. This polygon is the vertex figure, more precise formal definitions can vary quite widely, according to circumstance. For example Coxeter varies his definition as convenient for the current area of discussion, most of the following definitions of a vertex figure apply equally well to infinite tilings, or space-filling tessellation with polytope cells. Make a slice through the corner of the polyhedron, cutting all the edges connected to the vertex. The cut surface is the vertex figure and this is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places, Wenninger cuts each edge a unit distance from the vertex, as does Coxeter. For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint, other authors make the cut through the vertex at the other end of each edge. For irregular polyhedra, these approaches may produce a figure that does not lie in a plane. A more general approach, valid for convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices. Cromwell makes a cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere, many combinatorial and computational approaches treat a vertex figure as the ordered set of points of all the neighboring vertices to the given vertex. In the theory of polytopes, the vertex figure at a given vertex V comprises all the elements which are incident on the vertex, edges, faces. More formally it is the -section Fn/V, where Fn is the greatest face and this set of elements is elsewhere known as a vertex star. A vertex figure for an n-polytope is an -polytope, for example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a 4-polytope is a polyhedron. Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two vertices from an original face

10.
Stellation
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In geometry, stellation is the process of extending a polygon, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. The new figure is a stellation of the original, the word stellation comes from the Latin stellātus, starred, which in turn comes from Latin stella, star. In 1619 Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron and he stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to obtain the stella octangula, stellating a regular polygon symmetrically creates a regular star polygon or polygonal compound. These polygons are characterised by the number of times m that the polygonal boundary winds around the centre of the figure, like all regular polygons, their vertices lie on a circle. M also corresponds to the number of vertices around the circle to get one end of a given edge to the other. A regular star polygon is represented by its Schläfli symbol, where n is the number of vertices, m is the used in sequencing the edges around it. Making m =1 gives the convex, if n and m do have a common divisor, then the figure is a regular compound. For example is the compound of two triangles or hexagram, while is a compound of two pentagrams. Some authors use the Schläfli symbol for such regular compounds, others regard the symbol as indicating a single path which is wound m times around n/m vertex points, such that one edge is superimposed upon another and each vertex point is visited m times. In this case a modified symbol may be used for the compound, a regular n-gon has /2 stellations if n is even, and /2 stellations if n is odd. Like the heptagon, the octagon also has two octagrammic stellations, one, being a star polygon, and the other, being the compound of two squares. A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound, the interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, for a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells - we say that the cells in such a congruent set are of the same type. A common method of finding stellations involves selecting one or more cell types and this can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way. A set of cells forming a layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types, based on such ideas, several restrictive categories of interest have been identified. Adding successive shells to the core leads to the set of main-line stellations

11.
Polyhedron model
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A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material. Polyhedron models are found in mathematics classrooms much as globes in geography classrooms, Polyhedron models are notable as three-dimensional proof-of-concepts of geometric theories. Some polyhedra also make great centerpieces, tree toppers, Holiday decorations, the Merkaba religious symbol, for example, is a stellated octahedron. Constructing large models offer challenges in engineering structural design, construction begins by choosing a size of the model, either the length of its edges or the height of the model. The size will dictate the material, the adhesive for edges, the construction time, a single-colour cardboard model is easiest to construct — and some models can be made by folding a pattern, called a net, from a single sheet of cardboard. Choosing colours requires geometric understanding of the polyhedron, one way is to colour each face differently. A second way is to all square faces the same, all pentagonal faces the same. A third way is to colour opposite faces the same, many polyhedra are also coloured such that no same-coloured faces touch each other along an edge or at a vertex. For example, a 20-face icosahedron can use twenty colours, one colour, ten colours, or five colours, an alternative way for polyhedral compound models is to use a different colour for each polyhedron component. One way is to copy templates from a book, such as Magnus Wenningers Polyhedron Models,1974. A second way is drawing faces on paper or with computer-aided design software, the exposed nets of the faces are then traced or printed on template material. A third way is using the software named Stella to print nets, a model, particularly a large one, may require another polyhedron as its inner structure or as a construction mold. A suitable inner structure prevents the model from collapsing from age or stress, the net templates are then replicated onto the material, matching carefully the chosen colours. Cardboard nets are usually cut with tabs on each edge, so the step for cardboard nets is to score each fold with a knife. Panelboard nets, on the hand, require molds and cement adhesives. Assembling multi-colour models is easier with a model of a simpler related polyhedron used as a colour guide, complex models, such as stellations, can have hundreds of polygons in their nets. Recent computer graphics technologies allow people to rotate 3D polyhedron models on a video screen in all three dimensions. Recent technologies even provide shadows and textures for a realistic effect